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Abstract:

In this paper we show that, in the aftermath of a currency crisis, a government that adjusts the nominal interest rate in response to domestic currency depreciation can induce aggregate instability in the economy by generating self-fulfilling endogenous cycles. We find that, if a government raises the interest rate proportionally more than an increase in currency depreciation, then it induces self-fulfilling cycles that, driven by people's expectations about depreciation, replicate several of the salient stylized facts of the "Sudden Stop" phenomenon. These facts include a decline in domestic production and aggregate demand, a collapse in asset prices, a sharp correction in the price of traded goods relative to non-traded goods, an improvement in the current account deficit, a moderately higher CPI-inflation, more rapid currency depreciation, and higher nominal interest rates. In this sense, an interest rate policy that responds to depreciation may have contributed to generating the dynamic cycles experienced by some economies in the aftermath of a currency crisis.

1 Introduction

One of the policy debates that emerged
out of the Asian Crisis of 1997 was that of the appropriate
interest rate policy to fight domestic currency depreciation. On
one hand, the IMF advocated for higher interest rates to prevent
excessive depreciation. It claimed that such a policy would reduce
capital outflows by raising the cost of currency speculation and
induce capital inflows by making domestic assets more
attractive.3 On the other hand, some critics of the
IMF policy prescription recommended lowering interest rates in the
face of depreciation.4 These critics argued that lower rates
would alleviate the subsequent recessions in the Asian economies,
and improve the banks' and corporations' balance sheets, which had
weakened significantly as a result of the negative effects of rapid
currency depreciation on their dollarized liabilities.

These two views had something in common. They both advocated for
changes in interest rates in response to some macroeconomic
indicators, in this case the currency depreciation rate. They both
implied a feedback mechanism, positive or negative, from the
current depreciation rate to the nominal interest rate.5 In this
paper, we study some of the possible macroeconomic consequences of
these interest rate feedback policies in countries that have been
hit by a currency crisis. Our main result is that in the aftermath
of a crisis, an interest rate policy that calls for changing the
nominal interest rate, positively or negatively, in response to
currency depreciation can induce aggregate instability in the
economy by generating self-fulfilling endogenous cycles. That is,
this policy can cause cycles in the economy that are driven
exclusively by people's self-fulfilling beliefs rather than
fundamentals, and regardless of whether the response to
depreciation is positive or negative.

We find that if a government raises the interest rate
proportionally more than an increase in currency depreciation, then
it induces self-fulfilling cycles that replicate several of the
salient empirical regularities of emerging market crises. In
particular, we build a model in which these cycles, which are
driven exclusively by people's expectations about currency
depreciation, replicate many of the stylized facts following a
currency crisis. These facts, labeled by Calvo (1998) as the "Sudden Stop" phenomenon, include a decline in domestic production
and aggregate demand, a collapse in asset prices, a sharp
correction in the price of traded goods relative to non-traded
goods, an improvement in the current account deficit, a moderately
higher CPI-inflation, more rapid currency depreciation and higher
nominal interest rates. In this sense, interest rate policies that
respond to depreciation may have contributed to generating the
dynamic cycles experienced by some economies in the aftermath of a
currency crisis.

As a building block to our full model, we first construct a
simple environment of a small open economy in which we are able to
derive analytical results. We show that interest rate
policies that respond only to currency depreciation can
induce real indeterminacy or multiple equilibria, which in turn
opens the possibility of self-fulfilling equilibria.6 The
intuition for this result is the following. Suppose that in
response to a "sunspot," agents in the economy expect a higher
inflation rate. Because the interest rate policy responds
only to currency depreciation - and not to inflation - the
real interest rate can fall, boosting aggregate demand. As firms
see aggregate demand rising, they raise prices, thus validating the
original expectations of a higher inflation.7

Our full model is a sticky-price small open economy with traded
and non-traded goods with two important features: an interest rate
policy that responds to currency depreciation, and a collateral
constraint which captures the fact that international loans must be
guaranteed by physical assets such as capital. Loosely speaking,
the crisis is modelled as a time when this constraint is
unexpectedly binding.8 The model also includes other features
that have become very useful to match quantitatively some of the
stylized facts of the aftermath of a crisis. In particular, we
consider non-traded distribution services, and domestic and
international loan requirements to hire labor and to purchase an
imported intermediate input.9

In this full model, an interest rate policy that responds only
to depreciation also induces multiple equilibria. However, the
interaction of this policy with the collateral constraint induces
self-fulfilling cycles. This result is related to, but
differs from, the seminal work by Kiyotaki and Moore (1997). They
show that a binding collateral constraint induces credit cycles
that may amplify business cycles driven by fundamentals. In our
work this constraint creates cycles, but the interest rate policy
makes them self-fulfilling to the extent that they are driven by
people's expectations about the economy.

We think our results have at least three important implications.
First, they do not support any of the views of the previously
mentioned debate. In fact we unveil a peril that may be present in
both policy recommendations. What seems crucial is not whether the
government increases or decreases the interest rate, but instead
the feedback response of the nominal interest rate to currency
depreciation that most of the previous studies have ignored.

Second, our results provide a possible explanation of why the
empirical literature has not been able to obtain conclusive
evidence about whether higher interest rates can cause nominal
currency depreciation or, instead, appreciation in the aftermath of
a crisis.10 This literature has tried to control
for the variables that influence the nominal exchange rate. But our
results suggest that there can be potential influences that may
depend on "sunspots," which in turn can induce self-fulfilling
cycles in the nominal exchange rate (or the nominal depreciation
rate) as well as in other variables. Clearly these influences do
not depend on fundamentals, and their effect should be taken into
account by this literature.

Third, since interest rate policies that respond to currency
depreciation can induce expectations-driven fluctuations, then they
can destabilize the economy. Therefore they can be costly in terms
of macroeconomic instability and welfare.11 This has not been
studied in the previous literature and deserves further research.
For instance, Lahiri and Vegh (2002, 2003) and Flood and Jeanne
(2000) focus on the fiscal and output costs of higher interest
rates before and after a crisis. In addition Lahiri and Vegh (2003)
claim that there is a non-monotonic relationship between welfare
and the increase in interest rates. Christiano, Gust and Roldos
(2004) explore conditions under which a cut (rise) in the interest
rate in the midst of a crisis will stimulate output and improve
welfare. Aghion, Bacchetta and Banerjee (2000) find that it might
not be optimal to raise interest rates either when the proportion
of foreign currency debt is not too large, or when credit
provision, domestic investment and production are highly sensitive
to changes in nominal interest rates; whereas Braggion, Christiano
and Roldos (2005) build a model where, in response to a financial
crisis, it is optimal to raise the interest rate immediately, and
then reduce it sharply.

The remainder of this paper is organized as follows. In Section
2, we consider a simple economy in "good" times and characterize
analytically the equilibrium under interest rate policies that
respond only to currency depreciation. In Section 3, we build our
full model, which includes, among other things, a collateral
constraint. Here we study the determinacy of equilibrium analysis
through numerical simulations. In addition, we show that even if
the interest rate policy responds to past CPI-inflation, it can
still induce multiple equilibria as long as it responds to currency
depreciation.12 In Section 4, we use our full model
to construct a self-fulfilling equilibrium that captures the
stylized facts of a "Sudden Stop." Section 5 concludes.

2 The Simple Model

In this section we develop a simple
infinite-horizon small open economy model. The economy is populated
by a continuum of identical household-firm units and a government
who are blessed with perfect foresight. Before we describe in
detail the behavior of these agents we state a few general
assumptions and definitions.

There are two consumption goods: a traded good and a composite
non-traded good whose prices are denoted by
and respectively. We assume that the
law of one price holds for the traded good. Then
where
is the nominal exchange rate,
and
is the foreign price of the
traded good. Below we will relax this assumption. We also set
implying that
.

The real exchange rate () is defined as
the ratio between the price of traded goods and the aggregate price
of non-traded goods,
. From this
definition we deduce that

(1)

where
is the
gross nominal depreciation, and
is
the gross non-traded goods inflation

3 The Government

The government issues two nominal
liabilities: money, , and a domestic
bond,
that pays a gross nominal interest
rate . It does not have access to foreign
debt and makes lump-sum transfers to the household-firm units,
pays interest on its domestic debt,
(
, and receives revenues from
seigniorage Its budget constraint is described by
where
and
It follows a Ricardian fiscal policy by setting the lump-sum
transfers in order to satisfy the intertemporal version of its
budget constraint and its transversality condition

The description of monetary policy is motivated by the debate
that emerged after the Asian crisis about the appropriate interest
rate policy to fight against currency depreciation. We assume the
government implements an interest rate policy that responds to the
deviation of the current depreciation rate from a depreciation
target.13 That is,

(2)

where is a continuous, differentiable, and
strictly positive function in its argument with
and
; and
and
are the targets of the
nominal interest rate and the nominal depreciation rate that the
government wants to achieve.14

This policy can respond positively to the deviation of the
nominal depreciation rate from its target,
, capturing the policy
recommendation of the IMF; or it can react negatively,
, describing, to some
extent, the policy recommendation of the opposite view.
Nevertheless we exclude the cases
15 In other words,
the interest rate policy corresponds to (2)
with
and
either
or

2.2 The Household-Firm Unit

There is a large number of identical
household-firm units. They have perfect foresight, live infinitely
and derive utility from consuming, not working, and liquidity
services of money. The intertemporal utility function of the
representative unit is described by

(3)

where
corresponds to the subjective
discount rate, and
denote the consumption of traded and non-traded goods respectively,
and
are the labor allocated to the
production of the traded good and the non-traded good, and
refers to real money holdings measured
with respect to foreign currency. The specification in (3) assumes separability in the single period utility
function among consumption, labor, and real money balances. Hence
there are no distortionary effects of transactions money
demand.16 Moreover the utility function is
separable in
and
This allows us to derive
analytical results in the determinacy of equilibrium
analysis. To complete the characterization we also make the
following assumption.

The household-firm unit produces a flexible-price traded good
and a sticky-price non-traded intermediate differentiated good by
employing labor from perfectly competitive markets. For simplicity
we assume that there is zero labor mobility.17 The technologies
are described by

and

where
and
denote the labor hired by
the household-firm unit for the production of the traded good and
the non-traded good, respectively. The technologies satisfy the
following assumption.

Consumption of the non-traded good,
is a composite good made of a
continuum of intermediate differentiated goods. The aggregator
function is of the Dixit-Stiglitz type. Each household-firm unit is
the monopolistic producer of one variety of non-traded intermediate
goods. The demand for the intermediate good is of the form
satisfying
and
with
where
denotes the level of aggregate demand for the non-traded good,
is the nominal price of
the intermediate non-traded good produced by the household-firm
unit, and is the price of the composite
non-traded good. The unit sets the price of the good it supplies,
, taking the level of
aggregate demand for the good as given. Specifically, the
monopolist is constrained to satisfy demand at that price. That
is,

(4)

Following Rotemberg (1982), we introduce nominal price
rigidities for the intermediate non-traded good. The household-firm
unit faces a resource cost of the type
where
is the steady-state level of
the gross non-traded goods inflation.

There are incomplete markets. The representative household-firm
unit has access to the following two risk free bonds: a government
domestic bond, , that pays a gross nominal
interest rate and a foreign bond,
that pays a gross foreign
interest rate
In addition, the unit receives
income from working,
,
transfers from the government, , and
dividends. Then the flow budget constraint in units of the traded
good can be written as

(5)

where
with and

(6)

Equation (5) says that the end-of-period real
financial domestic assets (money plus domestic bond) can be worth
no more than the real value of financial domestic wealth, brought
into the period, plus the sum of wage income, transfers, and
dividends (
) net of consumption. The
dividends described in (6) correspond to the
difference between sale revenues and costs, taking into account
that through the firm-side the representative unit can hold foreign
debt,
. For holdings of foreign debt
the unit pays interests (
Moreover
the unit is subject to a Non-Ponzi game condition

(7)

where

The problem of the household-firm unit consists of choosing the
set of sequences
in order to maximize
(3) subject to (4),
(5), (6) and (7), given the initial condition
and the set of sequences {
, ,
, ,
Note that since the utility
function specified in (3) implies that the
preferences of the agent display non-sasiation, then both
constraints (5) and (7) hold
with equality. The Appendix contains a detailed derivation of the
necessary conditions for optimization. Imposing these conditions
along with the market clearing conditions in the labor markets, the
equilibrium symmetry (
and
the market
clearing condition for the non-traded good

(8)

and the definitions
and
we obtain

(9)

(10)

(11)

(12)

(13)

where
corresponds to the marginal cost of producing the non-traded good.
In addition equilibrium in the traded good market implies that

(14)

The interpretation of these equations is straightforward.
Condition (9) corresponds to an Uncovered
Interest Parity condition (UIP) that equalizes the returns of the
foreign and domestic bonds. Equation (10) makes the marginal rate of substitution
between labor, assigned to the production of the traded good, and
consumption of the traded good equal to the marginal product of
labor in the production of the traded good. Equations (11) and (12) are the standard
Euler equations for consumption of the traded good and consumption
of the non-traded good. Equation (13)
corresponds to the augmented Phillips curve for the sticky-price
non-traded goods inflation. And (14) describes
the dynamics of the current account deficit.

2.3 Capital Markets

We introduce imperfect international capital markets using the following ad-hoc supply curve of
funds

with

(15)

and where
corresponds to the country-specific risk premium, and is the risk free international interest rate. This
specification captures the idea that the small borrowing economy
faces a world interest rate,
, that increases when the stock
of foreign debt,
, is above its long run level,
. Then as the external debt
grows, so does the risk of default, and in order to compensate the
lenders for this risk, the economy has to pay them a premium over
the risk free international interest rate. We also assume that the
long-run level of foreign stock of debt is positive.

Assumption 3.The long-run level of the foreign stock
of debt is positive:

By introducing (15), we "close the small
open economy" and avoid the unit root problem, as discussed in
Schmitt-Grohé and Uribe (2003). This will allow us to pursue
a meaningful analysis of the dynamics of the non-linear equations
that describe the economy, using a log-linear
approximation.18

2.4 A Perfect Foresight Equilibrium

We are ready to provide a definition of
a perfect foresight equilibrium in this economy.

Definition 1Given the initial
condition the steady-state level of
foreign debt
and the depreciation
target a symmetric perfect
foresight equilibrium is defined as a set of sequencessatisfying:
a) the market clearing conditions for the non-traded and traded
goods, (8) and (14), b)
the UIP condition (9), c) the intratemporal
efficient condition (10), d) the Euler
equations for consumption of traded and non-traded goods, (11) and (12 ), e) the
augmented Phillips curve, (13), f) the
interest rate policy (2) and g) the ad-hoc
upward-sloping supply curve of foreign funds (15).

Since fiscal policy is Ricardian, this definition ignores the
budget constraint of the government and its transversality
condition. The definition also ignores real money balances. This is
because monetary policy is described as an interest rate policy in
(2),and real balances enter in the utility
function in a separable way. In addition, note that once we solve
for
it is
possible to retrieve the set of sequences
using (1), (5), and equations (41)-(44), (46), and (48) that are
presented in the Appendix.

2.5 The Determinacy of Equilibrium Analysis

In order to pursue the determinacy of
equilibrium analysis, we log-linearize the system of equations that
describe the dynamics of this economy around a steady state
. In the Appendix we
characterize this steady state.

Log-linearizing the equations of Definition 1 around the
steady-state yields

with
and either
or

(16)

with

(17)

(18)

(19)

(20)

(21)

where
and

(22)

whose signs are derived using Assumptions 1, 2 and 3. Equations
(16 )-(21)
correspond to the reduced log-linear representations of the
interest rate policy, the UIP condition, the Euler equation for
consumption of the non-traded good, the augmented Phillips curve,
the current account equation, and the Euler equation for
consumption of the traded good, respectively.

Our main goal is to show that the policy in (16) is prone to induce multiple equilibria in the
economy described by equations (17
)-(21). Proving this implies that this
policy can cause fluctuations in the economy that are driven by
people's self-fulfilling beliefs rather than fundamentals. In fact,
before we provide a formal proof, it is worth developing a simple
intuition. To do so, it is sufficient to concentrate on equations
(16)-(19). Note
that given the international interest rate,
then the policy
(16) and the UIP condition (17) determine the dynamics of the depreciation rate,
and the nominal interest
rate,
Moreover, the nominal interest
rate,
is not affected by either the
non-traded good inflation,
or the consumption of
the non-traded good, . Taking this into
account, we can construct the following self-fulfilling
equilibrium.

Assume that agents, in response to a "sunspot," expect
a higher non-traded good inflation in the next period. Since the
interest rate policy does not react to these expectations, then the
real interest rate measured with respect to the expected non-traded
good inflation,
declines.19 In response, households increase
consumption of non-traded goods - see (18) - which leads firms to raise their prices,
inducing a higher non-traded inflation - see (19). But by doing this, firms end validating the
original non-traded inflation expectations.

This simple intuition is appealing but incomplete, unless we
show that all the equilibrium conditions (16)-(21) are satisfied on
the entire equilibrium path. In other words, we need to
characterize formally the equilibrium of this economy. To
accomplish this, we start by writing equations (16)-(21) as

(23)

Then, we use this system to find and to compare the dimension of
its unstable subspace with the number of non-predetermined
variables.20 If the dimension of this subspace is
smaller than the number of non-predetermined variables, then, from
the results by Blanchard and Kahn (1980), we can infer that there
exist multiple perfect foresight equilibria. This forms the basis
for the existence of self-fulfilling fluctuations.

The following Proposition states the main result of the
determinacy of equilibrium analysis: an interest rate policy that
raises or lowers the nominal interest rate in response to current
currency depreciation leads to multiple perfect foresight
equilibria or, equivalently, to real indeterminacy.

Proposition 1If the government follows an interest rate policy that
responds to currency depreciation such as
with
and either
or
then there exists a continuum of perfect foresight equilibria
(indeterminacy) in which the sequences
converge
asymptotically to the steady state. In addition,

Proof. The eigenvalues of the matrix in (23) correspond to
the roots of the characteristic equation
Using
the definition of in (23), this equation can be written as

(24)

where

By Lemma 4 in the Appendix,
has real roots
satisfying
,
and
The fifth
root of
is
Clearly, if
then
whereas
if
then
Therefore, using this, the characterization of the roots of
and (24), we can conclude the following. If
then
has three explosive
roots, namely
,
and
While if
then
has two explosive
roots, namely
and
Hence,
regardless of whether
or
there are at most three explosive roots. Given that there are four
non-predetermined variables,
,
,
and
, then the number of
non-predetermined variables is greater than the number of explosive
roots. Applying the results of Blanchard and Kahn (1980), it
follows that there exists an infinite number of perfect foresight
equilibria converging to the steady state. Finally, parts a) and b)
follow from the difference between the number of non-predetermined
variables and the number of explosive roots, when
and
respectively.

Proposition 1 has two important
implications. First, provided that the fiscal policy is Ricardian,
then the interest rate policy induces not only real indeterminacy
but also nominal indeterminacy. That is, the nominal exchange rate
is not pinned-down.22 This follows from the definition of
the nominal depreciation rate, and the fact that the nominal
exchange rate is also a non-predetermined variable. Second, and in
contrast to the intuition provided above, Proposition 1 suggests that it is possible to construct
self-fulfilling equilibria that are based on expectations of a
different variable from the non-traded inflation. For instance, we
can construct a self-fulfilling equilibrium driven by people's
beliefs about currency depreciation. We will pursue this exercise
in Section 4.

Note: refers to multiple equilibria and refers to a unique equilibrium.

The results of Proposition 1 also pose the
question of whether policies that respond exclusively to either the
future depreciation rate, a forward-looking policy
or to the
past depreciation rate, a backward-looking policy
, can induce
multiple equilibria. The answer is yes and the analysis is provided
in the Appendix. In particular, we find that forward-looking
policies can still lead to multiple equilibria when the interest
rate response coefficient to future depreciation satisfies either
or
On the
contrary, backward-looking policies that are very aggressive with
respect to past depreciation, and satisfy
will guarantee a unique equilibrium; whereas timid policies that
satisfy
can lead to real indeterminacy. These results, as well as the
results from Proposition 1, are presented in
Table 1 in the columns labeled as "The Simple Model." 23

Note that the assumption of zero labor mobility is not crucial
for the results of Proposition 1. In fact, it
is possible to obtain similar analytical results under perfect
labor mobility and some extra assumptions about the disutility of
working.24 Then, what are the features of the
model that drive these results? The crucial features are the
following: the description of monetary policy as an interest rate
feedback rule, non-traded goods price-stickiness, and the exclusive
dependence of the policy on currency depreciation. By Sargent and
Wallace (1975), we know that the first feature by itself leads to
nominal indeterminacy of the exchange rate level (price level), in
a flexible price model under a Ricardian fiscal policy. The second
characteristic together with the policy elucidate why nominal
indeterminacy turns into real indeterminacy. And finally, the first
two features, in tandem with the exclusive response of the policy
to depreciation, are what explains why multiple equilibria arise,
to some extent, regardless of the degree of responsiveness of the
policy.25

This simple model suffers from at least two drawbacks. On one
hand, there is no feature that captures the fact that the economy
is in a crisis. On the other hand, the dynamics of non-traded
consumption and inflation, that are supported as a self-fulfilling
equilibrium, are completely at odds with the stylized facts of a "Sudden Stop." In particular, the self-fulfilling equilibrium that
we constructed above implies that non-traded consumption and
inflation are positively correlated. In contrast, the stylized
facts suggest that there was a strong decline in consumption
accompanied by an increase in inflation. To improve the model, in
the next section, we will add some extra features.

3 The Full Model

In this section we enrich the simple
model in several dimensions. First, in accord with Burnstein,
Eichenbaum, and Rebelo (2005a,b), we introduce non-traded
distribution services. This together with price stickiness are
crucial to explain the large movements in real exchange rates after
large devaluations. Second, following Lahiri and Vegh (2002) and
Christiano, Gust, and Roldos (2004) we assume that the
household-firm unit requires domestic and international loans to
hire labor and to purchase an imported intermediate input,
respectively. This is important to obtain a decline in output and
demand in the midst of the crisis, when interest rates rise. Third,
as in the new literature about currency crises, we introduce a
collateral constraint. That is, international loans must be
guaranteed by physical assets such as capital. This will help to
provide a definition of a crisis. Fourth, we consider a utility
function that is not separable in the two types of consumption. We
proceed to explain how we introduce these features and their
influence in the previous equations.

3.1 The Additional Features

As in Burnstein et al. (2003), we
assume that the traded good needs to be combined with some
non-traded distribution services before it is consumed. In order to
consume one unit of the traded good, it is required units of the basket of differentiated non-traded goods.
Let
and be the producer's price of the
traded good, the consumer's price of the traded good, and the
general price level of the basket of differentiated non-traded
goods, respectively. All of them are expressed in domestic
currency. Hence the consumer's price is simply
And since PPP holds at the production level of the traded good (
), and
the foreign price of the traded good is normalized to one (
), we have that

The production of the differentiated non-traded good is still
demand determined by

(25)

where
denotes the level of aggregate
demand for the non-traded good,
is the price of the
intermediate non-traded good set by the household-firm unit, and
corresponds to the level of
aggregate consumption of the traded good. But now the demand
requirements come from two sources.26They come from
consumption of non-traded goods
that provide utility, and from non-traded distribution services
that are necessary to bring one unit of the traded good to the
household-firm unit. Note that there is no difference between
non-traded consumption goods and non-traded distribution services.
As a consequence, in equilibrium the basket of non-traded goods
required to distribute traded goods will have the same composition
as the non-traded basket consumed by the household-firm unit.

The introduction of the loan requirements and the collateral
constraint in the model follows Christiano et al. (2004). The
household-firm unit requires domestic loans to hire labor (
and
), and international loans
to buy an imported input (), that will be
used in the production of the traded good. These loans are obtained
at the beginning of the period and repaid at the end of the period.
In this sense, they represent short-term debt and differ from
long-term foreign debt
We do not model, however,
the financial institutions that provide these loans. Instead, we
assume that the domestic loans are provided by the government,
whereas foreign loans are supplied by foreign creditors.27 For
these loans the unit pays interests
and (
that
are accrued between periods, where is the
domestic nominal interest rate, and is
the international interest rate. The latter is assumed to be
constant and equal to
.

The production technology of the traded good uses labor (
), an imported input
(), and capital ();
whereas the technology for the non-traded intermediate
differentiated goods only requires labor (
) and capital (). That is,

and

Furthermore, as in Christiano et al. (2004) and Mendoza and Smith
(2002), among others, capital is assumed to be time-invariant, does
not depreciate, and there is no technology to making it bigger.
Under these new features, the dividends that the household-firm
unit receives can be written as

To model the crisis, we follow Christiano et al. (2004) by
imposing the following collateral constraint:

(27)

where and
represent the real value, in units of foreign currency, of one unit
of capital for the production of the non-traded and traded goods,
respectively, and is the fraction of these
stocks that foreign creditors accept as collateral. The constraint
(27) says that the total value of foreign
and domestic debt, that the representative household-firm unit has
to pay to completely eliminate the debt of the firm by the end of
period cannot exceed the value of the
collateral. The crisis makes this constraint unexpectedly binding
in every period henceforth without the possibility of being
removed.

Finally, we relax the assumption in (3) of a
separable utility function between the two types of consumption,
. But we will still
assume separability among consumption, labor, and real money
balances. We proceed to study how all these features influence the
previous optimal conditions.

3.2 The New Equilibrium
Conditions

The problem that the household firm
unit has to solve is similar to the one presented in the simple
model. The agent chooses the set of sequences
in order to maximize

subject to the budget constraint

and the constraints (7), (25), (26), and (27), given the initial conditions
and
and the set of sequences {
, ,
,
The optimization conditions
together with symmetry and market clearing conditions can be used
to find the laws of motion of the economy. They correspond to
(1), (2), (27) with equality,

(28)

(29)

(30)

where

(31)

(32)

(33)

(34)

(35)

(36)

(37)

(38)

where
and
are the Lagrange multipliers of
the collateral constraint and the budget constraint. The latter
multiplier evolves according to the asset pricing equation

(39)

Equations (28)-(34)
are equivalent to equations (8)-(14) in the simple model.28 Therefore they have a
similar interpretation. Equations (35)-(37) correspond to the
optimal conditions that determine the demands for labor (for the
production of the traded and non-traded good) and for the imported
input; and (38) is the intratemporal
condition that makes the marginal rate of substitution between
labor, assigned to the production of the non-traded good, and
consumption of the non-traded good equal to the real salary
measured in units of the non-traded good.

The new laws of motion reveal that the additional features have
some important consequences. On one hand, distribution services
affect the relative price of the traded good at the consumer level
with respect to the nominal exchange rate. In the simple model this
relative price was equal to one. In the full model this price is
which depends on the
distribution costs parameter From (30) and (31), it is clear
that through this price, distribution costs affect, in equilibrium,
the optimal intratemporal decisions between labor and consumption
of the traded good, as well as the optimal intertemporal choices
for consumption of the traded good. On the other hand, distribution
services generate an extra demand of non-traded goods, as is
captured by the last term,
, of the right hand side of
(28). This extra demand also influences
the dynamics of non-traded goods inflation, as can be seen in
(33).

The binding collateral constraint generates an endogenous risk
premium, as reflected by the modified UIP condition in (29). In fact, because of this constraint, the
effective international interest rate that domestic agents pay
becomes
Thus, raising
external debt
not only requires the payment
of interests (
), but also tightens
the binding constraint (
). This generates an
additional interest cost.29

The requirement of loans to hire labor and the binding
collateral constraint affect the labor demand decisions, as can be
inferred from (35) and (36). Keeping the rest constant, these two features and
the fact that in the short run
imply the following.
Increases in the effective interest rate,
push the cost of hiring
labor up, which in turn discourages the demands for labor for the
production of the traded and non-traded goods.

The optimal condition for demand of the imported input, equation
(37), equalizes the marginal product of this
input to the effective cost of foreign working capital,
that is necessary to
import it. As the constraint tightens and
goes up, the effective cost raises and the demand for the imported
input decreases. Furthermore, the purchases of this input influence
the current account equation (34). A decrease
in the imported input can cause an improvement in the current
account deficit. This improvement is almost immediate, given the
assumption that external short term loans to finance the
intermediate input have to be repaid at the end of the period, and
not at the beginning of next period.

Finally, it is possible to derive the equilibrium value of the
prices of capital. The equilibrium value of these asset prices are
described by

and

(40)

where and are the
marginal products of capital in the production of the traded and
non-traded goods, respectively.

3.3 The Determinacy of Equilibrium
Analysis

Using this set-up, the definition of
equilibrium under a currency crisis is the following.

Definition 2Given,
and the
depreciation target a perfect foresight
equilibrium is defined as a set of sequences,
,
satisfying equations (1), (2), the collateral constraint (27) as an equality, (28)-(37) and (40).

Note that, as in Christiano et al. (2004) among others, we model
the crisis as a collateral constraint that binds in every period.
Nevertheless, at the steady state, the shadow price of the
collateral constraint is equal to zero. To see this use (39) in tandem with
to obtain
This implies that the
collateral constraint is marginally not binding at the
steady state. Hence the credit restrictions disappear marginally.
In contrast, in the short run the shadow price of the collateral
constraint may vary as changes. When
is high then the collateral
constraint tightens.

As before, to pursue the determinacy of equilibrium analysis, we
log-linearize the system of equations around the perfect-foresight
steady state. Then we characterize the dimension of the unstable
subspace of the system and compare it to the number of
non-predetermined variables. By log-linearizing the system, we are
following the same approach that Kiyotaki and Moore (1997) adopt to
solve for an equilibrium of a model with a binding collateral
constraint. This precludes the possibility of exploring non-linear
equilibrium dynamics.30Moreover, since it is not possible to
derive analytical results, we have to rely on numerical
simulations. We use the following functional forms. For consumption
and labor preferences we use

and

where
,
a and
whereas for technologies we
utilize

and

where
and

Table 2.

1.06

0.943

1.16

17.5

6

0.85

0.185

0.7

a

0.4

2

4.59

5

0.6

1.4

3.5

0.5

0.7

0.64

1

2

The values of the parameters are mainly borrowed from the
calibration by Christiano et al. (2004), except for the
intratemporal elasticity of substitution a) the parameter related to distribution
services (, the parameter that governs the
degree of price stickiness ( and the
parameter associated with the degree of imperfect competition
(31 We do not pick any particular
value for the interest rate response coefficient to currency
depreciation (
), since we will study how
this parameter affects the determinacy of equilibrium. We choose
values for "a" and that
are in line with similar values used in the distribution services
literature.32 Since there are no robust estimates
of a New-Keynesian Phillips curve for emerging economies, we choose
values for and that are
consistent with the values used in the closed economy literature
about nominal price rigidities.33 Table 2 summarizes
the parametrization.

Using this parametrization, we study how the determinacy of
equilibrium varies with respect to the response coefficient to
depreciation (
) of the policy (2) and other structural parameters. As an illustrative
case, we focus on the experiment of varying the intratemporal
elasticity of substitution (a), while keeping the other structural
parameters constant. The results are presented in Figure 1, where a
cross "x" denotes combinations of
and "a" under which the
policy induces multiple cyclical equilibria, whose degree of
indeterminacy is of order one; and a dot "." represents parameter
combinations under which the policy induces multiple cyclical
equilibria, whose degree of indeterminacy is of order two.34 As
can be seen in this figure, a policy that responds to current
currency depreciation, by raising (
with
) or lowering (
with
) the nominal interest
rate, can induce multiple equilibria regardless of the
intratemporal elasticity of substitution "a."

Figure 1. Contemporaneous Policies

Figure 1: Characterization of the equilibrium for interest rate
policies
varying the degree of responsiveness to currency
depreciation (
) and
the intratemporal elasticity of substitution (a). It is assumed
that
. A cross "x" denotes parameter combinations under which
the policy induces multiple cyclical equilibria whose degree of
indeterminacy is of order one. A dot "." represents parameter
combinations under which the policy induces multiple cyclical
equilibria whose degree of indeterminacy is of order
two.

Figure 1 - Detailed Description: Figure 1 shows the characterization of
the equilibrium for contemporaneous interest rate policies
-.5in0pt-.5in0pt-.5in0pt-.5in0pt-.5in0pt-.5in0pt-.5in1046pt-.5in1396pt varying
the degree of responsiveness to currency depreciation (
) and the
intratemporal elasticity of substitution (a). The figure measures
on the
vertical axis with
and "a" on the horizontal axis with a
The figure
shows that for a
, any interest
policy that satisfies
will induce multiple cyclical equilibria whose degree of
indeterminacy is of order one. In addition for a
, any interest
policy that satisfies
will induce multiple
cyclical equilibria whose degree of indeterminacy is of order two.
For
and a
the interest
rate policy will lead to multiple equilibria, but depending on the
value of "a" the order of indeterminacy could be either one or
two.

Experiments with respect to other structural parameters lead to
similar results.35 This suggests that the results in
the full set-up are similar to the ones in the simple set-up. But
there is an important distinction. Now because of the binding
collateral constraint, there exist self-fulfilling cycles
or, equivalently, multiple cyclical equilibria. This is just
a consequence of two mechanisms working together. On one hand, from
the results in the simple model we have that this policy can induce
self-fulfilling "non-cyclical" fluctuations. On the other hand,
from Kiyotaki and Moore (1997) we know that the introduction of a
binding collateral constraint can cause credit cycles. Hence
the combination of the two mechanisms can lead to self-fulfilling
cyclical equilibria. The following Proposition summarizes
these results.

Proposition
2Under a currency crisis, if the
government follows an interest rate policy that responds to
currency depreciation, such as
with
and either
or
then there
exists a continuum of perfect foresight cyclical equilibria
(indeterminacy), in which the sequences,
converge to
the steady state.

These results are not specific to the particular policy that we
consider. In the Appendix we study forward-looking policies,
and
backward-looking policies,
. We find
that forward-looking policies always induce multiple cyclical
equilibria, as long as the response coefficient to future
depreciation satisfies either
or
.
Except for the presence of cycles these results coincide, to some
extent, with the ones from the simple model. On the other hand, for
backward-looking policies the coefficient of response to past
depreciation plays an important role in the characterization of the
equilibrium. That is, timid rules satisfying
always induce multiple equilibria, while aggressive rules with
can guarantee a unique equilibrium. Nevertheless, being aggressive
with respect to past depreciation (
)
is a necessary but not a sufficient condition to guarantee a unique
equilibrium. Therefore, backward-looking policies can still
destabilize the economy by inducing self-fulfilling cyclical
fluctuations.

These results, as well as the results for a contemporaneous
policy
are
summarized in Table 1 in the columns labeled as "The Full
Model." Note that our results do not support any of the views of
the debate about the right interest rate policy in the aftermath of
the Asian crisis. In fact, we unveil a peril that may be present in
both policy recommendations. What is crucial in our analysis is not
whether the government increases or decreases the interest rate;
instead, it is the feedback response of the nominal interest rate
to current, future, or past currency depreciation, that most of the
previous studies have ignored.

It is possible to argue that, in the aftermath of a crisis,
governments may also adjust the nominal interest rate in response
to inflation. This raises the question of whether a policy that
reacts to both the CPI-inflation and the depreciation rate can
still induce aggregate instability in the economy. The answer to
this question is affirmative making our previous results stronger.
To some extent, the response to currency depreciation is crucial to
explain the possibility of multiple equilibria. To see this, we can
study the following interest rate policy

with and
either or

that reacts, positively or negatively, to current currency
depreciation and, aggressively and positively, to past
CPI-inflation.36

Figure 2. Responding to Past CPI Inflation and Current Depreciation

Figure 2: Characterization of the equilibrium for the rule
varying the degrees of responsiveness to past
CPI-inflation (
) and to
current currency depreciation (
). It is
assumed that
. A cross "x" denotes parameter combinations associated
with multiple equilibria whose degree of indeterminacy is of order
one. These equilibria can be cyclical or non-cyclical. A dot "."
represents parameter combinations associated with multiple
equilibria whose degree of indeterminacy is of order two. These
equilibria are cyclical and we name these combinations as "MCE(2)". The white regions represent parameter combinations under
which there exists a unique equilibrium.

Figure 2 - Detailed Description:
Figure 2 shows the characterization of
the equilibrium for the rule
varying the degrees of responsiveness to past CPI-inflation
(
) and to
current currency depreciation (
). The
figure measures
on the
vertical axis with
and
on the
horizontal axis with
The figure shows that for
,
any interest policy that satisfies
will induce multiple equilibria (cyclical and non-cyclical) whose
degree of indeterminacy is of order one. For the region within the
coordinates (
and approximately, the rule will
induce multiple equilibria (cyclical and non-cyclical) whose degree
of indeterminacy is either of order one or two. For the region
within the coordinates (
and approximately, the rule will
lead to a unique equilibrium.

The results of the analysis are shown in Figure 2. We study how
variations of the degrees of responsiveness to past CPI-inflation
and current depreciation,
and
, affect the characterization
of the equilibrium. We observe that policies that respond to both
past CPI-inflation and currency depreciation can still
induce multiple equilibria. In fact, using the parametrization in
Table 2 and under the celebrated "Taylor coefficient",
any policy that responds to
currency depreciation will lead to real indeterminacy. Note also
that if the policy is positively aggressive with respect to current
depreciation,
, then excessively
aggressive policies with respect to inflation, say any
will continue to
induce self-fulfilling equilibria. Nevertheless, this analysis also
highlights the importance of reacting aggressively to past
CPI-inflation in order to avoid destabilizing the economy.

There is still a relevant question that has not been answered.
If the governments of the Asian economies followed these interest
rate policies, then would it be possible to support the stylized
facts of the aftermath of the crisis, the "Sudden Stop"
phenomenon, as one of these self-fulfilling equilibria? An
affirmative answer to this question would make our previous results
more credible. This defines the goal of the next section.

4 Constructing a Self-fulfilling Cyclical Equilibrium

In this section, we use the full model
in tandem with an interest rate policy that responds to currency
depreciation, in order to construct a self-fulfilling cyclical
equilibrium that replicates some of the stylized facts of the "Sudden Stop" phenomenon. In this equilibrium, the only source of
cyclical fluctuations is the self-validation of people's beliefs
about currency depreciation.

We assume the government follows an interest rate policy,
, that responds
aggressively,
,
to current depreciation,
.37 This policy
captures the immediate reaction to currency depreciation.
Unfortunately, in the empirical literature that emerged after the
Asian crisis, there are no robust estimates for the
parameter
. For illustrative purposes
we set
implying that the
government increases the interest rate proportionally more than the
increase in depreciation.

From Proposition 2 we know that varying
will not preclude the
possibility of multiple equilibria. This, however, changes the
degree of indeterminacy. For
the degree of
indeterminacy is one.38 Then we can construct a
self-fulfilling cyclical equilibrium in which a "sunspot"
affects people's expectations of only one variable of the
economy, such as currency depreciation39
Moreover, as long as
, increasing or reducing
will not change the
qualitative results that we will present, and that capture some of
the stylized facts of the "Sudden Stops."

We assume the crisis occurs exogenously. That is, the binding
collateral constraint is exogenously imposed at time as in Christiano et al. (2004). In this sense, we are
only interested in studying what happens in the economy at and in
the aftermath of the crisis. In what follows, therefore, we
concentrate exclusively in the equilibrium dynamics of the economy
at and after

Imagine that when the crisis hits the economy at time
people develop expectations, in
response to a "sunspot", of a 10% higher domestic currency
nominal depreciation. By the determinacy of equilibrium analysis,
we know that these expectations will be self-validated. In
addition, they induce cycles as described by the impulse response
functions presented in Figures 3 and 4. In these figures, all the
variables but the multiplier of the collateral constraint are
measured as percentage deviations from the steady state. A quick
inspection reveals that at time and for some
subsequent periods, the self-fulfilling equilibrium captures the
following stylized facts of the "Sudden Stops": a decline in the
aggregate demand (consumptions of traded and non-traded goods and
aggregate consumption), a collapse in the domestic production
(traded output and non-traded output), a collapse in asset prices
(prices of traded and non-traded capital), a sharp correction in
the price of traded goods relative to non-traded goods, an
improvement in the current account deficit, a moderately higher
CPI-inflation, a more rapid currency depreciation, and higher
nominal interest rates. After some periods, domestic output and
aggregate demand increase, the current account deficit
deteriorates, and CPI-inflation, currency depreciation and interest
rates decrease. Then, the cycles are quickly dampened, and the
economy converges to the steady state.

We proceed to explain these results. The self-validated increase
in the nominal depreciation rate implies an increase in the nominal
exchange rate and, by the policy, leads to an increase in the
nominal interest rate at . Since the rule is
aggressive, the nominal interest rate rises by more than both the
expected non-traded good inflation rate and the expected traded
good inflation rate at time .40
Hence, the real interest rate at measured in
terms of either the expected traded inflation or the non-traded
inflation at goes up. Provided that this induces
an intertemporal substitution effect in consumption that more than
offsets any intratemporal substitution effect, then consumption of
the traded good and consumption of the non-traded good decline at
. This can be inferred from (31) and (32). As a result of
this, aggregate consumption also decreases implying that the model
is able to capture the initial decline in aggregate demand present
in the "Sudden Stops."

Figure 3. Impulse Responses

Figure 3: Impulse responses of a self-fulfilling equilibrium, when at
time people expect a
higher nominal depreciation rate (
This equilibrium replicates some of the "Sudden Stops"
stylized facts: a decline of consumption of traded and non-traded
goods, a relatively moderate increase in CPI-inflation, a higher
depreciation rate, and a higher nominal interest rate. All the
variables are measured as percentage deviations from the steady
state except for the multiplier.

Figure 3 - Detailed Description:
Figure 3 presents the impulse responses
of a self-fulfilling equilibrium, when at time (5.1, -3) people expect a higher nominal
depreciation rate (t = 0). The figure has 9 panels that
correspond to the impulse responses for the nominal depreciation
rate (
), the
nominal interest rate (), the non-traded inflation rate (
), the
traded inflation rate (
), the
CPI-inflation rate (), the traded good consumption (), the non-traded good
consumption (),
the aggregate consumption (), and the multiplier (
).
All the variables are measured as percentage deviations from the
steady state except for the multiplier. In general the impulse
responses replicate some of the "Sudden Stops" stylized facts: a
decline of consumption of traded and non-traded goods, a relatively
moderate increase in CPI-inflation, a higher depreciation rate, and
a higher nominal interest rate. We proceed to describe these
responses individually.

At (5.1, -3) the
percentage deviation of the nominal depreciation rate jumps to . After this, it
declines and converges to zero over time. It reaches the value of
zero when 10% approximately.

At (5.1, -3) the
percentage deviation of the nominal interest rate jumps to t = 5. After this, it
declines and converges to zero over time. It reaches the value of
zero when 10% approximately.

At (5.1, -3) the
percentage deviation of the non-traded inflation rate jumps to 20% approximately. After this,
it declines and converges to zero over time. It reaches the value
of zero when 17.5% approximately.

At (5.1, -3) the
percentage deviation of the traded inflation rate jumps to t=7 approximately. After this,
it declines and converges to zero over time. It reaches the value
of zero when 17.5% approximately.

At (5.1, -3) the
percentage deviation of the CPI-inflation rate jumps to 15% approximately. After this,
it declines and converges to zero over time. It reaches the value
of zero when 17.5% approximately.

At (5.1, -3) the
percentage deviation of the traded good consumption reaches the
value of 16% approximately. At -5% it
attains a maximum of t = 4 approximately and then it converges to zero over time. It reaches
the value of zero when 0.4% approximately.

At (5.1, -3) the
percentage deviation of the non-traded good consumption reaches the
value of t = 9 approximately. At -0.9% it
attains a maximum of t = 2 approximately and then it converges to zero over time. It reaches
the value of zero when 1.4% approximately.

At (5.1, -3) the
percentage deviation of the aggregate consumption reaches the value
of t = 8 approximately.
At -3.8 it attains a
maximum of t = 3 approximately and then it converges to zero over time. It reaches
the value of zero when 1.4 % approximately.

At (5.1, -3) the
multiplier reaches the value of 0.5% and then declines and converges to zero over
time. It reaches the value of zero when 0.095 approximately.

Figure 4. Impulse Responses

Figure 4: Impulse responses of a self-fulfilling equilibrium, when at
time people expect a
higher nominal depreciation rate (
This equilibrium replicates some of the "Sudden Stops"
stylized facts: a collapse in the domestic production (of the
traded and non-traded good), a collapse in asset prices (prices of
capital), a sharp correction in the price of traded goods relative
to non-traded goods, and an improvement in the current account
deficit. All the variables are measured as percentage deviations
from the steady state.

Figure 4 - Detailed Description:
Figure 4 presents the impulse responses
of a self-fulfilling equilibrium, when at time (5.1, -3) people expect a higher nominal
depreciation rate t = 0. The figure has 9 panels that
correspond to the impulse responses for the effective nominal
interest rate (t = 6),
the imported input (),
the marginal cost of producing the non-traded good (), the traded output
(), the
non-traded output (), the current account deficit, the price of
traded goods relative to non-traded goods, the price of capital
allocated to the production of the traded good (
) and the
price of capital allocated to the production of the non-traded good
(
.

All the variables are measured as percentage deviations from the
steady state. In general the impulse responses replicate some of
the "Sudden Stops" stylized facts: a collapse in the domestic
production (of the traded and non-traded good), a collapse in asset
prices (prices of capital), a sharp correction in the price of
traded goods relative to non-traded goods, and an improvement in
the current account deficit. We proceed to describe the impulse
responses individually.

At (5.1, -3) the
percentage deviation of the effective interest rate jumps to 17.5% approximately.
After this, it declines and converges to zero over time. It reaches
the value of zero when 0.095 approximately.

At (5.1, -3) the
percentage deviation of the imported input jumps to 38%. After this, it converges to
zero over time reaching the value of zero when 0.095 approximately.

At (5.1, -3) the
percentage deviation of the marginal cost jumps to -21% approximately. After this, it declines and
converges to zero over time. It reaches the value of zero when 0.095 approximately.

At (5.1, -3) the
percentage deviation of the traded output jumps down to 21% approximately. After this,
it converges to zero over time, reaching the value of zero when 17.5% approximately.

At (5.1, -3) the
percentage deviation of the non-traded output jumps down to -9% approximately 17.5%. At -3.8% it attains a
maximum of -3.5% approximately and then it converges to zero over time. It reaches
the value of zero when 1.4% approximately.

At (5.1, -3) the
percentage deviation of the current account reaches the value of 0.7% approximately.
At -3.8% it attains a
maximum of -2.3% approximately and then it converges to zero over time. It reaches
the value of zero when 1.4% approximately.

At (5.1, -3) the
percentage deviation of the price of traded goods relative to
non-traded goods jumps to 0.6% approximately. After this, it converges to zero
over time reaching the value of zero when 0.095 approximately.

At (5.1, -3) the
percentage deviation of the price of capital allocated to the
production of the traded good jumps down to 32% approximately. After this, it converges to zero
over time reaching the value of zero when 10% approximately.

At (5.1, -3) the
percentage deviation of the price of capital allocated to the
production of the non-traded good reaches the value of -23% approximately. At 10% it attains a maximum of -18% approximately and
then it converges to zero over time. It reaches the value of zero
when 1.4% approximately.

Since the real value of capital as a collateral is expressed in
terms of foreign currency, then the previously mentioned increase
of the nominal exchange rate at t = 0 reduces this
value. Thus, there is an incentive to reduce the demand for the
imported input at t = 0, as can be deduced from
(27) as an equality, and taking into
account that
is a predetermined variable.
At the same time, a higher interest rate in response to
depreciation also pushes up the costs of loans to hire labor
utilized in the production of traded goods. This creates an
incentive for the household-firm unit to cut back labor in the
production of the traded good.41 The simulations show that in fact
these incentives are materialized leading to a decrease in traded
output at t = 0.

On the other hand, the supply of the non-traded good is demand
determined. This supply satisfies both consumption of non-traded
goods and distribution services for traded goods. Consequently, the
previously mentioned decrease in demand of both goods causes a
decrease in non-traded output (labor) at t = 0. Thus
the model is able to capture the decline in output of both
non-traded and traded goods. Provided that the decrease in traded
output is smaller than the contractions in consumption of the
traded good and the imports of the intermediate input, then the
current account improves, as can be seen in (34).

As the collateral constraint tightens and the nominal interest
rate rises, the "effective" nominal interest rate
increases. As a result of
this, marginal costs of producing the non-traded good go up,
forcing the household-firm unit to raise the price of this good.
This, in turn, leads to a higher non-traded goods inflation rate,
as can be deduced from (33) and (36). Higher currency depreciation rates and non-traded
inflation rates induce higher CPI-inflation rates.

In addition, the presence of price-stickiness and distribution
services implies that, as a consequence of large depreciations,
there is a sharp correction in the price of the traded good
relative to the price of non-traded good at the consumer level
.
This captures another empirical regularity of the aftermath of a
crisis.

The real value of a unit of capital for traded output
(respectively, non-traded output), in terms of foreign currency, is
determined by the net present value of the flows of the marginal
product of capital in the production of the traded good
(respectively, non-traded good). This can be seen by iterating
forward equations (40). Provided that
capital is constant in the analysis, the decreases in labor and in
the intermediate input cause a decline in the marginal product of
capital, reducing the real value of capital. A similar mechanism
reduces the real value of capital for non-traded output. Therefore,
asset prices fall capturing another stylized fact of the "Sudden
Stops."

5 Concluding Remarks

In this paper we study interest rate
policies that, in the aftermath of a currency crisis, call for
adjusting the nominal interest rate in response to domestic
currency depreciation. We show that these policies can induce
macroeconomic instability by generating self-fulfilling cycles. We
find that, if a government raises the interest rate proportionally
more than an increase in currency depreciation, then it induces
self-fulfilling cycles that, driven by people's expectations about
depreciation, replicate several of the salient stylized facts of
the "Sudden Stop" phenomenon. In this sense, interest rate
policies that respond to depreciation may have contributed to
generating the dynamic cycles experienced by some economies in the
aftermath of a currency crisis.

Our results have the following implications. Previous works have
emphasized that these interest rate policies can cause fiscal and
output costs. We suggest that these policies can be also costly to
the extent that they can induce macroeconomic instability in the
economy by opening the possibility of "sunspot" equilibria.
These equilibria, that are not driven by fundamentals, can be
associated with a large degree of volatility for some macroeconomic
aggregates such as consumption. Provided that agents are risk
averse, then volatile consumption can be costly in terms of
welfare.

Our results also provide a possible explanation of why the
empirical literature has not been able to disentangle the
relationship between interest rates and the nominal exchange
(depreciation) rate in the aftermath of a crisis. This literature
has tried to control for the variables that influence the nominal
exchange rate. But our results suggest that there can be potential
influences that may depend on "sunspots", which in turn can
induce self-fulfilling cycles in the nominal exchange rate (or the
nominal depreciation rate) as well as in other variables. Clearly,
these influences do not depend on fundamentals, and their effect is
something that the empirical literature should take into
account.

An issue that we have not addressed is whether the "sunspot"
equilibria are learnable by the agents of the economy. It was
implicitly assumed that agents could learn and coordinate their
actions on any particular equilibrium. In future research, we plan
to relax this assumption and use the Expectational Stability
concept developed in Evans and Honkapojha (2001) to pursue a
learnability analysis.

A Appendix

This Appendix has two parts. The first
part includes material that supports the analysis for the simple
model of Section 2. The second part includes the simulations of the
determinacy of equilibrium analysis for forward-looking and
backward-looking rules for the full model of Section 3.

A.1 The Simple Model

A.1.1 The First Order Conditions of the
Household-Firm Unit Problem in the Simple Model

The representative household-firm unit
chooses the set of sequences
in order to maximize
(3) subject to (4),
(5), (6) and (7), given the initial condition
and the set of sequences {
, ,
,
The first order conditions
correspond to (5) and (7)
with equality and

where
and
correspond to the Lagrange
multipliers of (4) and (5)
respectively.

We will focus on a symmetric equilibrium in which all the
monopolistic producers of sticky-price non-traded goods pick the
same price. Hence
Since all the
monopolists face the same wage rate,
and the same production function,
then they will
demand the same amount of labor
In equilibrium
the money market, the domestic bond market, the labor markets, the
non-traded goods market, and the traded good market clear. In
particular, for the non-traded good market and traded good market
we have

(51)

and

(52)

Combining (49) and (50) yields the uncovered interest parity condition
(9). From (43) and
(45) we can derive equation (10). Using conditions (41)
and (49) we obtain the Euler equation for
consumption of the traded good that corresponds to (11). Utilizing (1), (41), (42), and (50) we derive the Euler equation (12 ) for consumption of the non-traded good. And
finally using the notion of a symmetric equilibrium, conditions
(4), (42), (44), (46), (47), the labor market equilibrium conditions, and
the definitions
and
we can derive the
augmented Phillips curve described by equation (13).

A.1.2 Characterization of the Steady State
in the Simple Model

We use
, (1),
(8 )-(15), and the
condition that at the steady state
for all the variables to
derive

Then it is simple to prove that under some assumptions that include
Assumptions 1, 2, and 3, and given and
there exists a steady
state
for the economy that
provided a particular
satisfies these equations
with
and
In particular to guarantee that there exist
we need
some Inada-type assumptions such as
and
42

A.1.3 Forward-Looking and Backward-Looking
Policies in the Simple Model

In order to pursue the determinacy of
equilibrium analysis for a forward-looking policy,
we use this
and equations (17)-(21) to obtain

(53)

Then the determinacy of equilibrium analysis delivers the results
stated in the following proposition.

Proposition 3If the government follows a forward-looking interest
rate policy such as
with
and either
or
then there exists a continuum of perfect foresight equilibria
(indeterminacy) in which the sequences
converge asymptotically to the steady state. The degree of
indeterminacy is of order 1.43

Proof. The eigenvalues of the matrix in (53) correspond to the roots
of the characteristic equation
By
Lemma 4 we know that
has real roots
satisfying
,
and
Therefore
has only two explosive
roots, which means that the matrix in
(53) has two explosive eigenvalues. Given
that there are three non-predetermined variables namely
,
and
, then the number of
non-predetermined variables is greater than the number of explosive
roots. Applying the results of Blanchard and Kahn (1980), it
follows that there exists an infinite number of perfect foresight
equilibria converging to the steady state. In addition, the
difference between the number of non-predetermined variables and
explosive roots implies that the degree of indeterminacy is of
order 1.

To derive the results for a backward-looking policy we use the
log-linearized version of this policy,
together with
equations (20)-(17) to
obtain the log-linearized system

(54)

Then the determinacy of equilibrium analysis delivers the results
stated in the following proposition.

Proposition 4Assume the government follows a backward-looking policy
such as
with

a) if
then there exists a unique perfect foresight equilibria in which
the sequences
converge to the steady state.

b) if
then there exists a continuum of perfect foresight equilibria
(indeterminacy) in which the sequences
converge
asymptotically to the steady state. In addition, the degree of
indeterminacy is of order 2.

Proof. The eigenvalues of the matrix in (54) correspond to the roots
of the characteristic equation
Using
the definition of in (54), this equation can be written as

(55)

where
is defined in Lemma
4. Using this Lemma we know that
has real roots
satisfying
,
and
The fifth
and the sixth roots of
are
and
Clearly, if
then
and
whereas
if
then
and
Using
this, the characterization of the roots of
and (55) we can conclude the following. If
then
has four explosive
roots, namely
,
,
and
While if
then
has two explosive
roots, namely
and
Therefore
if
the number of explosive roots is equal to the number of
non-predetermined variables (
,
,
and
Hence, applying the
results of Blanchard and Kahn (1980), it follows that there exists
a unique equilibrium. This completes the proof for a).

On the contrary, by the previous analysis, if
then the number of explosive roots, 2, is less than the number of
non-predetermined variables (
,
,
and
Applying the results of Blanchard and Kahn (1980), it follows that
there exists an infinite number of perfect foresight equilibria
converging to the steady state. The degree of indeterminacy is the
difference between the number of non-predetermined variables and
the number of explosive roots. This completes the proof for b).

A.1.4 Lemmata and Proofs for the Results in
the Simple Model

Proof. First recall from Azariadis (1993) that a
sufficient condition to have real roots is that
. To prove a)note that
means that
But this implies
that which in turn leads to
. Hence the
roots are real. Next we prove b).
means that
But this implies
that that in turn leads to
. Hence the
roots are real.

Lemma 2Define
The roots of the characteristic equation
are real and
satisfy
and

Proof. First using the definition of
Assumptions 1, 2
and 3 and definitions in (22) we obtain
that
and
Since
then by Lemma
1 we know that the two roots are real. In
addition, from Azariadis (1993), having
and
imply that one
root lies inside of the unit circle and the other one lies outside
the unit circle. Without loss of generality we can conclude that
and

Lemma 3Define
The roots of the characteristic equation
are real and
satisfy
and

Proof. First using the definition of
Assumptions
1, 2 and 3 and definitions in (22) we
obtain that:
and
Since
then by
Lemma 1 we know that the two roots of
are real. In
addition, from Azariadis (1993), having
and
imply
that one root lies inside the unit circle and the other one lies
outside the unit circle. Without loss of generality we can conclude
that
and

Proof. By Lemma 2 we know that
has two real roots
satisfying
and
On the
other hand, by Lemma 3, we know that
has two real
roots satisfying
and
Using
these and (56) the result of the
Lemma follows.

A.2 The Full Model

A.2.1 Forward-Looking and Backward-Looking
Policies in the Full Model

We characterize the equilibrium for
forward-looking policies (
) and backward
looking policies (
), using the
parametrization of Table 2. As an illustrative case, we focus on
the experiment of varying the degree of responsiveness to future
(past) currency depreciation (
) and the intratemporal
elasticity of substitution (a), keeping the rest constant. The
results are presented in Figure 5. The top panel shows the results
for a forward-looking policy. The bottom panel presents the results
for a backward-looking policy.

Figure 5. Forward-Looking Policies

Figure 5: Characterization of the equilibrium for forward-looking
(top-panel) and backward-looking (bottom-panel) policies varying
the degree of responsiveness to currency depreciation (
) and
the intratemporal elasticity of substitution (a). It is assumed
that
. A cross "x" denotes parameter combinations under which
the policy induces multiple equilibria whose degree of
indeterminacy is of order one. A dot "." represents parameter
combinations under which the policy induces multiple equilibria
whose degree of indeterminacy is of order two. The white regions
represent parameter combinations under which there exists a unique
equilibrium.

The top panel (forward-looking policies) measures
0.9% the
response coefficient to future depreciation, on the vertical axis
with
and "a" on the horizontal axis with a
The panel
shows that for a
, any
interest policy that satisfies
will induce multiple cyclical equilibria whose degree of
indeterminacy is of order one.

The bottom panel (backward-looking policies) measures
0.9% the
response coefficient to past depreciation, on the vertical axis
with
and "a" on the horizontal axis with a
The panel
shows that for a
, any
interest policy that satisfies
will induce multiple
equilibria (cyclical or non-cyclical) whose degree of indeterminacy
is of order two. For
then depending on a
the
backward-looking policy can induce either a unique equilibrium or
multiple cyclical equilibria whose degree of indeterminacy is of
order one. In particular in the region within the coordinates a,
(0, 3) and (2.5, 3), approximately the policy
always induces a unique equilibrium. In addition for
then depending on a
the
backward-looking policy can induce either a unique equilibrium or
multiple cyclical equilibria whose degree of indeterminacy is of
order two. In particular in the region within the coordinates a,
(0, 3), and (2.5, -3), approximately the policy always induces a unique equilibrium.

From the top-panel, we can infer that forward-looking policies
always induce multiple cyclical equilibria, as long as
and either
or
. On the other
hand, for backward-looking rules, the coefficient of response to
past depreciation,
plays an important role. In
particular, timid rules with respect to past depreciation (
)
always induce multiple equilibria, regardless of the intratemporal
elasticity of substitution (a). In contrast, aggressive rules (
)
can guarantee a unique equilibrium. Nevertheless, being aggressive
with respect to past depreciation (
)
is not a sufficient condition to guarantee a unique equilibrium. It
is only a necessary condition.

Varying other structural parameters different from the
intratemporal elasticity of substitution (a), in tandem with
lead to similar
results.44 The following proposition summarizes
these results.

Proposition 5Under a currency
crisis,

If the government follows a forward-looking policy such as
with
and either
or
then there exists a continuum of perfect foresight "cyclical"
equilibria (indeterminacy), in which the sequences,
converge to
the steady state.

If the government follows a backward-looking policy such as
with
then

a)
is a sufficient condition for the existence of a continuum of
perfect foresight equilibria, possibly "cyclical", in which the
sequences,
converge to
the steady state.

b)
is a necessary but not a sufficient condition for the existence of
a unique perfect foresight equilibrium, where the sequences,
converge to
the steady state.

Footnotes

1. This paper is based on chapter 3 of my
dissertation at the University of Pennsylvania. I am grateful to
Martín Uribe, Stephanie Schmitt-Grohé and Frank
Schorfheide for their guidance and teaching. I also received
helpful suggestions and benefitted from conversations with Roc
Armenter, Martin Bodenstein, David Bowman, Sanjay Chugh, Bill
Dupor, Bora Durdu, Fabio Ghironi, Chris Gust, Dale Henderson,
Sylvain Leduc, Gustavo Suárez and seminar participants at
the International Economics Brown Bag Lunch at the University of
Pennsylvania, the International Finance Workshop at the Federal
Reserve Board and the fall 2005 SCIEA Meetings. All errors remain
mine. Previous versions of this paper circulated under the title:
"Interest Rate Rules and Multiple Equilibria in the Aftermath of
BOP Crises". The views expressed in this paper are solely the
responsibility of the author and should not be interpreted as
reflecting the view of the Board of Governors of the Federal
Reserve System or of any other person associated to the Federal
Reserve System. Return to text

4. See for instance Furman and Stiglitz
(1998) and Radelet and Sachs (1998) among others. Return to text

5. In fact some works motivated by the
debate describe, implicitly or explicitly, the interest rate policy
as a feedback rule responding to some measure of nominal
depreciation. See for instance Cho and West (2001), Goldfajn and
Baig (1998), and Lahiri and Vegh (2003), among
others. Return to text

6. From now on we will use the terms "multiple equilibria" and "real indeterminacy" (a "unique
equilibrium" and "real determinacy") interchangeably. By real
indeterminacy we mean that the behavior of one or more (real)
variables of the economy is not pinned down by the model. This
implies that there are multiple equilibria, which in turn opens the
possibility of having fluctuations in the economy generated by
endogenous beliefs that are of the "sunspot" type; i.e., they
are based on stochastic variables that are extrinsic in Cass and
Shell's (1983) terminology. Return to
text

7. An interest rate policy that responds to
inflation does not necessarily preclude the possibility of
self-fulfilling equilibria. But as we discuss below, the response
to currency depreciation makes the interest rate policy more prone
to induce self-fulfilling equilibria. Return to text

8. This idea of modelling the crisis as an
unexpected binding collateral constraint captures the essence of a
"Sudden Stop." Other works that introduce such collateral
constraints include Caballero and Krishnamurthy (2001), Christiano,
Gust and Roldos (2004), Krugman (1999), Mendoza and Smith (2002),
and Paasche (2001) among others. Return
to text

9. See Burnstein, Eichenbaum and Rebelo
(2005a,b) and Christiano, Gust and Roldos (2004), among
others. Return to text

11. These policies can lead to "sunspot"
equilibria that are characterized by a large degree of volatility
of some macroeconomic aggregates such as consumption. Provided that
agents are risk averse, then these policies can induce equilibria
where agents are worse-off. Return to
text

12. The new interest rate rules
literature argues that aggressive rules with respect to past
inflation are more likely to guarantee a unique equilibrium. See
for instance Benhabib, Schmitt-Grohé, and Uribe (2001),
Taylor (1999), and Woodford (2003) among others. See also Zanna
(2003) for an analysis of interest rate rules in small open
economies. Return to text

13. Some of the works inspired by this
debate, such as Cho and West (2001), Goldfajn and Baig (1998), and
Lahiri and Vegh (2003) among others, also describe the interest
rate policy, implicitly or explicitly, as a rule that reacts to
some measure of nominal depreciation. Return to text

14. For simplicity we also assume that
these targets correspond to the steady-state levels of these
variables. Return to text

15. As we will see below, the cases of
or
introduce a unit root in
the log-linearized system of the economy, precluding the
possibility of using the "Theorem of Hartman and Grobman" to
derive meaningful conclusions about the dynamics of the non-linear
system. See Guckenheimer and Holmes (1985). Return to text

16. Because of this, we can write the
real money balances that enter the utility function in terms of
foreign currency,
without consequences for our results. Return to text

17. As we discuss below, our results do
not depend on this assumption. Return
to text

18. The "unit-root problem" that is
commonly present in small open economy models arises because of
assuming that
To see why,
use this assumption together with condition (49) to deduce that
This is an
equation that has a unit root and that introduces a unit root in
the entire dynamical system of the simple set-up. See
Schmitt-Grohé and Uribe (2003). Return to text

20. The dimension of the unstable
subspace is given by the number of roots of the system that are
outside the unit circle. See Blanchard and Kahn
(1980). Return to text

21. The degree of indeterminacy is
defined as the difference between the number of non-predetermined
variables and the dimension of the unstable subspace of the
log-linearized system. Return to
text

22. A Non-Ricardian fiscal policy
combined with the monetary policy under study will determine the
level of the nominal exchange rate if
but not if
. Return to text

23. Our general results still hold if we
describe monetary policy as
with and
This resembles the implicit
descriptions in some of the empirical works mentioned in Montiel
(2003). Return to text

24. When labor is mobile across sectors
then
and the equilibrium in
the labor market becomes
In
addition, if we assume that and
and keep the rest of Assumptions
1, 2, and 3, then the log-linearized system of equations that
describes the economy is still (16)-(21 ). But in this
case,
and
Return to text

25. In Zanna (2003), we show that in
order to guarantee a unique equilibrium, a rule must respond
aggressively to the non-traded inflation but timidly to
depreciation. A rule that responds aggressively to currency
depreciation still opens the possibility of multiple equilibria,
regardless of its response to non-traded inflation. Return to text

27. To formalize this point, we could
introduce financial institutions in the model that behave in a
perfectly competitive way and supply the aforementioned loans. This
would not change our main results. Return to text

29. Note also that in contrast to the
simple model, we have assumed that
. Nevertheless, in
this context, this typical assumption of the small open economy
literature does not cause the unit-root problem. Under this
assumption and with the binding constraint, condition (39) becomes
which does not introduce a unit root in the system of equations
that describes the economy. Return to
text

30. To pursue a determinacy of
equilibrium analysis in the augmented non-linear model is a
very challenging task. Most of the works, that include a collateral
constraint and that simulate equilibrium dynamics for the
non-linear system, do not characterize the equilibrium. They assume
that the equilibrium that is found computationally is the relevant
one, whose properties must be studied. See Mendoza and Smith (2002)
and Christiano et al. (2004), among others. Return to text

31. Note that target nominal depreciation
rate,
can be found by evaluating
(29) at the steady state. That is,
The
value of that we take is close to the one in
Christiano et al. (2004). Return to
text

36. The literature of interest rate rules
claims that an aggressive backward-looking rule with respect to
inflation is more prone to guarantee a unique equilibrium than
forward-looking and contemporaneous policies. See Woodford
(2003). Return to text

37. It is also possible to construct
self-fulfilling equilibria with the forward-looking and
backward-looking policies that, in principle, replicate most of the
stylized facts. Return to text

38. That is, in this case the dynamic
log-linearized system that describes the economy has complex and
non-explosive eigenvalues, and the number of non-predetermined
variables exceeds the number of explosive eigenvalues by
one. Return to text

39. If the degree of indeterminacy were
2, then we would have an extra degree of freedom. We could assume
that a "sunspot" affects the expectations of an additional
variable different from the depreciation rate. In this sense, we
are being conservative. Return to
text

40. This is probably the case because
there is sluggish price adjustment for the price of the non-traded
good that, in turn, affects the price of the traded good through
the existence of non-traded distribution services. Return to text

41. This can be deduced from (35) and the production technology of the traded
good. Labor costs increase not only because the nominal interest
rate increases but also because the collateral constraint tightens.
That is, there is an increase in the "effective" nominal
interest rate
. Return to text